2025/10/22 by Damian Sobota, Sobota, Damian, Tomasz Żuchowski +1 · 1 citation
Computer Science · Mathematics · #Advanced Algebra and Logic #Advanced Banach Space Theory #Advanced Topology and Set Theory #FOS: Mathematics #Functional Analysis (math.FA) #General Topology (math.GN) #Logic (math.LO)
paper · pdf · doi:10.48550/arxiv.2510.19744
openalex publication_date 2025/10/22 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28
For an ideal I in a σ-complete Boolean algebra A, we show that if the Boolean algebra A\langleI⟩ generated by I does not have the Nikodym property, then it does not have the Grothendieck property either. The converse however does not hold -- we construct a family of \mathfrakc many pairwise non-isomorphic Boolean subalgebras of the power set \wp(ω) of the form \wp(ω)\langleI⟩ which, when thought of as subsets of the Cantor space 2ω, belong to the Borel class \mathbbFσδ and have the Nikodym property but not the Grothendieck property, and a family of 2^\mathfrakc many pairwise non-isomorphic non-analytic Boolean algebras of the form \wp(ω)\langleI⟩ with the Nikodym property but without the Grothendieck property. Extending a result of Hernández-Hernández and Hrušák, we show that for an analytic P-ideal I on ω the following are equivalent: 1) I is totally bounded, 2) I has the Local-to-Global Boundedness Property for submeasures, 3) \wp(ω)/I contains a countable splitting family, 4) conv≤KI. Moreover, proving a conjecture of Drewnowski, Florencio, and Paúl, we present examples of analytic P-ideals on ω with the Nikodym property but without the Local-to-Global Boundedness Property for submeasures (and so not totally bounded). Exploiting a construction of Alon, Drewnowski, and Łuczak, we also describe a family of \mathfrakc many pairwise non-isomorphic ideals on ω, induced by sequences of Kneser hypergraphs, which all have the Nikodym property but not the Nested Partition Property -- this answers a question of Stuart. Finally, Tukey reducibility of a class of ideals without the Nikodym property is studied.